STOP Early Journal Content on JSTOR, Free to Anyone in the World This article is one of nearly 500,000 scholarly works digitized and made freely available to everyone in the world by JSTOR. Known as the Early Journal Content, this set of works include research articles, news, letters, and other writings published in more than 200 of the oldest leading academic journals. The works date from the mid-seventeenth to the early twentieth centuries. We encourage people to read and share the Early Journal Content openly and to tell others that this resource exists. People may post this content online or redistribute in any way for non-commercial purposes. Read more about Early Journal Content at http://about.jstor.org/participate-jstor/individuals/early- journal-content . JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. 410 BIOLOGY: A. J. LOTKA Proc. N. A. S. McEwen, G. F. and Michael, E. L. The functional relation of one variable to each of a number of correlated variables determined by a method of successive approxima- tion to group averages; a contribution to statistical methods. Proc. Amer. Acad. Arts Set., 55, 1919 (95-133). Reed, H. S. and Holland, R. H. The growth rate of an annual plant, Helianthus. Proc. Nat. Acad. Set., 5, 1919 (135-144). Robertson, T. B. On the nature of the autocatalyst of growth. Arch. Entwichl- mech., 37, 1913 (497-508). 1 Paper No. 64, University of California, Graduate School of Tropical Agriculture and Citrus Experiment Station, Riverside, California. 2 The writer wishes to acknowledge his great indebtedness to Dr. G. F. McEwen of the Scripps Institution for Biological Research of the University of California for valuable assistance in the mathematical work here reported. ANALYTICAL NOTE ON CERTAIN RHYTHMIC RELATIONS IN ORGANIC SYSTEMS By Alfred J. Lotka Brooklyn, N. Y. Communicated by R. Pearl, May 20, 1920 Periodic phenomena play an important r61e in nature, both organic and inorganic. In chemical reactions rhythmic effects have been observed experi- mentally, and have also been shown, by the writer 1 and others, 2 to follow, under certain conditions, from the laws of chemical dynamics. However, in the cases hitherto considered on the basis of chemical dynamics, the oscillations were found to be of the damped kind, and therefore, only transitory (unlike certain experimentally observed periodic reactions). Furthermore, in a much more general investigation by the writer, covering the kinetics not only of chemical but also of biological systems, it appeared, from the nature of the solution obtained, improbable 3 that undamped, permanent oscillations would arise in the absence of geometrical, structural causes, .in the very comprehensive class of systems considered. For it seemed that the occurrence of such permanent oscilla- tions, the occurrence of purely imaginary exponents in the exponential series solution presented, would demand peculiar and very specific rela- tions between the characteristic constants of the systems undergoing transformation; whereas in nature these constants would, presumably, stand in random relation. It was, therefore, with considerable surprise that the writer, on apply- ing his method to certain special cases, found these to lead to undamped, and hence indefinitely continued, oscillations. As the matter presents several features of interest, and illustrates certain methods and principles, it appears worth while to set forth the argument and conclusions here. Vol. 6, 1920 BIOLOGY: A. J. LOTKA 411 Starting out first from a broad basis, we may consider a system in the process of evolution, such a system comprising a variety of species of matter Si, S 2 . . . .S n of mass X\, Xi. . . .X n . The species of matter S may be defined in any suitable way. Some of them may, for example, be biological species of organism, others may be components of the "in- organic environment." Or, the species of matter S may be several com- ponents of an inorganic system in the course of chemical transformation. We may think of the state of the system at an instant of time as being defined by statement of the values of X\, X$. . . .X„; of certain para- meters Q defining the character of each species (in general, variable with time); and of certain other parameters P. The parameters P will, in general, define the geometrical constraints of the system, both at the boundaries (volume, area, extension in space), and also in its interior (structure, topography, geography); they will further define such factors as temperature and climatic conditions. For a very broad class of cases, including those commonly treated in chemical dynamics, but extending far beyond the scope of that branch of science, the course of events in such a system will be represented by a system of differential equations of the form §'-*<*.*. X n ;P,Q)) (1) (« = 1, 2. . . .n) ) If we restrict ourselves to the consideration of evolution at constant P's and Q's we may write briefly <¥f = Fi(X u X*. . . .XJ. (2) The writer has elsewhere 4 given a somewhat detailed discussion of the general case, in which no special assumption is made regarding the form of the functions F, that is to say, regarding the mode of physical inter- dependence of the several species and their environment. We now proceed to consider a simple special case, as follows : The system comprises 1. A species of organism Si, a plant species, say, deriving its nourish- ment from a source presented in such large excess that the mass of the source may be considered constant during the period of time with which we are concerned. 2. A species S2, for example a herbivorous animal species, feeding on Si. In this case we have the following obvious relations f Other dead Mass of Si or excretory destroyed by matter elimi- S2 per unit of nated from Si time per unit of I time J Rate of crease of per unit time m- Xi of Mass of newly = formed Si per i — unit of time (3) 412 BIOLOGY A. J. LOTKA Proc. N. A. S. Mass of newly ' Rate of in- formed S% per Mass of S 2 crease of X% per unit of = -1 unit of time (derived from - destroyed or eliminated per (4) time Si as food in- gested) unit of time Or, in analytical symbols, d -§ = A\X 1 -B 1 X 1 X i -A" 1 X 1 at (5) = (A\-A" 1 )X 1 -B 1 X 1 X i (6) = AiXj — -B1A1A2 (7) = X 1 (A 1 -B 1 X 2 ) (8) —r- = A 2X1X2 — BiXl at (9) = X,(A,Xi -£2) (10) The coefficients A\, An, Bj, B% are in general functions of X\ and X%. The reasons for selecting the form (5), (9) for the analytical formula- tion of (3), (4) require perhaps a little explanation. For small changes the rate of formation of new material of a given species of organism under given conditions is proportional to the existing mass of that species. In other words, the growth of living matter is a typically autocatakinetic 5 process. This term has, therefore, been put in the form A' X\ for the species Si. Proportionality does not hold for large changes of X\, X%, and this is duly provided for in that A '1 is a function of X\, X%. Similarly the mass of matter rejected per unit of time from the species Si is proportional to X\, and has been put in the form A'\X\, where A" is in general a function of X\, Xt. Again, the mass of Si destroyed by S 2 feeding upon it will, for small changes, be proportional to Xt and also to X\. This term has, therefore, been set down in the form B 1X1X2. Here again the departures from pro- portionality are taken care of by the variations of Bi with Xi and Xt, of which variables B\ is a function. Similar remarks apply to the formulation (9) of (4). If there were no waste in the feeding process, and assuming that Si consumes no other substance than Si, we would have B\ — A 2 ; but in the more general case B\ ± A 2 . Approaching now the analytical treatment of equations (5), (9), or their equivalents (8), (10), we note first of all that there are two ways of satisfying the condition for equilibrium, namely: Xi = X 2 = (11) and X 1 = h A 2 ' Xi = B x (12) Vol. 6, 1920 BIOLOGY: A. J. LOTKA 413 We shall return later to the condition (11). Condition (12) we will employ to define a new origin. Accordingly we introduce into (8), (10) the variables: (13) (14) and obtain d%\ It dx?. It Xi = Xi- "A 2 %2 = x 2 - Ai ~B X = 012*2 4- a m %i%2 = 021*1 4- 0212*1*2 (15) where 012 = — - (16) (17) (18) (19) B\B% a7 din = — Bi AtAz On- -g- 0212 = ^2 Note the significant fact that in (15) the linear terms in the dexter diagonal are lacking. It is this circumstance which imparts an oscillatory •character to the process. For, since #12 and 021 are in general functions of *i, x% let us expand them by Taylor's theorem and put an = po + piXi + P2X2 + (20) 021 = qo 4- 01*1 4- 92*2 4- (21) A general solution of the system of differential equations (15) is then *i = Pie Xlt 4- Pte M + Pue 2M + P^e 2M + * 2 = Qxe Xlt + Qj* + Q u e 2M 4- Q^e 2M + .... where Xi, X2, are the roots of the determinental equation for X -X po qo —X (23) (24) (25) that is to say, ^ = *><p q (26) Now, according to (20), (21) p Q , q are the equilibrium values of 012, .021. Hence, if we denote by Ai, Bi the equilibrium values of A h B 2 , i.e., those values which correspond to *i = * 2 = 0, then we have, by (16), .(18) poqo = -A1B2 (27) and hence X = ±V-AiS2 (28) 414 BIOLOGY: A. J. LOTKA Proc. N. A. S.. Now the coefficient 5 2 is, in the nature of things, a positive number, as follows from its definition by (4), (9). As regards the coefficient A\, we have two possible alternatives. If A i is negative for all values of X h X 2 , then X, as denned by (28), would be real; but this inference is nugatory. For B\, like B 2 is, by definition (3), (5), an essentially positive quantity, and hence the equilib- rium defined by (12) would in this case occur at a negative value of Xi. But this is physically impossible, since Xi is a mass. By referring to (5), (7) it will be seen that this case, in which A\ is negative for all values of X h X% and in which an equilibrium of the type defined by (12) is physically impossible, corresponds to a species Si in- capable of maintaining itself even in the absence of the tax placed upon it by the species S 2 feeding upon it. This is a case of minor interest. If, on the contrary (12) can be satisfied by a positive value of A\, so that an equilibrium of the type (12) is physically possible, then, evidently, by (28), X is a pure imaginary. The solution (23), (24) then takes the form of Fourier's series; the process is an undamped oscillation con- tinuing indefinitely. In this connection, it is interesting to recall a passage in Spencer's "First Principles," chapter 22, paragraph 173: "Every species of plant and animal is perpetually undergoing a rhyth- mical variation in number — now from abundance of food and absence of enemies rising above its average, and then by a consequent scarcity of food and abundance of enemies being depressed below its average amid these oscillations produced by their conflict, lies that average number of the species at which its expansive tendency is in equilibrium with surrounding repressive tendencies. Nor can it be questioned that this balancing of the preservative and destructive forces which we see going on in every race must necessarily go on. Since increase of numbers cannot but continue until increase of mortality stops it, and decrease of numbers cannot but continue until it is either arrested by fertility or extinguishes the race entirely." A question now arises. Do the curves representing the solution (23) , (24) ■ dip below the zero axes of X u Xi? This would mean that one or the other, or both, of the species Si, Si would become extinct through the violence of the oscillations. To answer this question we consider the relation: dXi _ Xt{AiX\—Bi) ,__. dXi ~ XlJAv^BA) which is obtained from (8) and (10) by division. From the periodicity of x\, xi (and, therefore, Xi, Xi) it follows that the curve defined in rectan- gular coordinates X u Xi by (29) is a closed curve. Furthermore, this curve can never cross the X\ axis, for at all points of this axis the first,. Vol. 6, 1920 BIOLOGY: A. J. LOTKA 415 and all the higher derivatives of Xi with regard to X\ vanish, as can be seen from (29) directly and by successive differentiations. Similarly it can be seen that the curve defined by (29) can never cross the Xi axis. Hence, if any point on any integral curve of (29) lies within the positive quadrant, the whole of that curve lies in that quadrant. Thus the oscillations can never exceed the limits of positive values Xi, Xz. We conclude, therefore, that under the conditions of the problem as here set forth, neither the species Si nor the species S2 can become extinct through severity of the oscillations alone. In practice the eventuality might arise, however, that in the course of these oscillations one or the other species might be so thinned out as to succumb to any extraneous influence that might arise such as has not been taken into account in our present considerations. We return now briefly to the consideration of the equilibrium defined by the equation Xt = X 2 = (11) Applying here the criterion set forth by the author elsewhere, 6 it is seen that when Ai is positive the determinental equation for X has at this point two real roots of opposite sign, which is characteristic of unstable equilib- rium. If, on the other hand, Ai is negative in the neighborhood of the origin of X lt X% then the equilibrium here is found to be stable, the two roots for X being both negative. In conclusion it may be remarked that a system of equations identical in form with (8), (10) is obtained in the discussion of certain consecutive autocatalytic chemical reactions. Here, however, the coefficients A, B are constants and the integration can be reduced to a quadrature. Aside from a certain number of periodic reactions which have been observed more or less as laboratory curiosities, a certain interest is also attached to this matter from the fact that rhythmical reactions (e.g., heartbeat, which may continue after excision), play an important rdle in physiology. We cannot, of course, say whether in such case geometrical (structural) features are the dominating factors. 1 Lotka, A. J., J. Phys. Chem., 14, 1910 (271-274); Zs. pkysik. Ckem., 72, 1910 (508- 511); 80, 1912 (159-164) ; Phys. Rev., 24, 1912 (235-238) ; Proc. Amer. Acad. Arts Sci., 55, 1920 (137-153). 2 Hirniak, J., Zs. physik. Chem., 75, 1910 (675); compare also Lowry and John, J. Chem. Soc, 97, 1910 (2634-2645). 3 Lotka, A. J., Proc. Amer. Acad., loc. cit., p. 145, footnote 13. 4 Lotka, A. J., Science Progress, 14, 1920 (406-417); Proc. Amer. Acad., loc. cit. 5 Ostwald, Wo., ttber die zeitlichen Eigenschaften der Entwickelungsvor gauge, Leipsic, 1908, p. 36. 6 Lotka, A. J., Proc. Amer. Acad., loc. cit., p. 144, et seq.